Simplify the following expression and state the condition under which the simplification is valid. You can assume that $p \neq 0$. $y = \dfrac{6p + 15}{7} \div \dfrac{2p^2 + 5p}{p} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{6p + 15}{7} \times \dfrac{p}{2p^2 + 5p} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ (6p + 15) \times p } { 7 \times (2p^2 + 5p) } $ $ y = \dfrac {p \times 3(2p + 5)} {7 \times p(2p + 5)} $ $ y = \dfrac{3p(2p + 5)}{7p(2p + 5)} $ We can cancel the $2p + 5$ so long as $2p + 5 \neq 0$ Therefore $p \neq -\dfrac{5}{2}$ $y = \dfrac{3p \cancel{(2p + 5})}{7p \cancel{(2p + 5)}} = \dfrac{3p}{7p} = \dfrac{3}{7} $